Triacontadigon

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Regular triacontadigon
Regular polygon 32.svg
A regular triacontadigon
Type Regular polygon
Edges and vertices 32
Schläfli symbol {32}, t{16}, tt{8}, ttt{4}
Coxeter–Dynkin diagrams CDel node 1.pngCDel 3x.pngCDel 2x.pngCDel node.png
CDel node 1.pngCDel 16.pngCDel node 1.png
Symmetry group Dihedral (D32), order 2×32
Internal angle (degrees)168.75°
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a triacontadigon (or triacontakaidigon) or 32-gon is a thirty-two-sided polygon. In Greek, the prefix triaconta- means 30 and di- means 2. The sum of any triacontadigon's interior angles is 5400 degrees.

Contents

An older name is tricontadoagon. [1] Another name is icosidodecagon, suggesting a (20 and 12)-gon, in parallel to the 32-faced icosidodecahedron, which has 20 triangles and 12 pentagons. [2]

Regular triacontadigon

The regular triacontadigon can be constructed as a truncated hexadecagon, t{16}, a twice-truncated octagon, tt{8}, and a thrice-truncated square. A truncated triacontadigon, t{32}, is a hexacontatetragon, {64}.

One interior angle in a regular triacontadigon is 1683⁄4°, meaning that one exterior angle would be 111⁄4°.

The area of a regular triacontadigon is (with t = edge length)

and its inradius is

The circumradius of a regular triacontadigon is

Construction

As 32 = 25 (a power of two), the regular triacontadigon is a constructible polygon. It can be constructed by an edge-bisection of a regular hexadecagon. [3]

Symmetry

Symmetries of triacontadigon.png The symmetries of a regular triacontadigon. Lines of reflections are blue through vertices, and purple through edges. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions.

The regular triacontadigon has Dih32 dihedral symmetry, order 64, represented by 32 lines of reflection. Dih32 has 5 dihedral subgroups: Dih16, Dih8, Dih4, Dih2 and Dih1 and 6 more cyclic symmetries: Z32, Z16, Z8, Z4, Z2, and Z1, with Zn representing π/n radian rotational symmetry.

On the regular triacontadigon, there are 17 distinct symmetries. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. [4] He gives r64 for the full reflective symmetry, Dih16, and a1 for no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular triacontadigons. Only the g32 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

32-gon with 480 rhombs
32-gon rhombic dissection-size2.svg
regular		 Isotoxal 32-gon rhombic dissection-size2.svg
Isotoxal

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. [5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular triacontadigon, m=16, and it can be divided into 120: 8 squares and 7 sets of 16 rhombs. This decomposition is based on a Petrie polygon projection of a 16-cube.

Examples
32-gon rhombic dissection.svg 32-gon rhombic dissection2.svg 32-gon rhombic dissectionx.svg 32-gon-dissection-random.svg

Triacontadigram

A triacontadigram is a 32-sided star polygon. There are seven regular forms given by Schläfli symbols {32/3}, {32/5}, {32/7}, {32/9}, {32/11}, {32/13}, and {32/15}, and eight compound star figures with the same vertex configuration.

Many isogonal triacontadigrams can also be constructed as deeper truncations of the regular hexadecagon {16} and hexadecagrams {16/3}, {16/5}, and {16/7}. These also create four quasitruncations: t{16/9} = {32/9}, t{16/11} = {32/11}, t{16/13} = {32/13}, and t{16/15} = {32/15}. Some of the isogonal triacontadigrams are depicted below as part of the aforementioned truncation sequences. [6]

Related Research Articles

Octagon Polygon shape with eight sides

In geometry, an octagon is an eight-sided polygon or 8-gon.

Icosagon Polygon with 20 edges

In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

Triacontagon Polygon with 30 edges

In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

Pentadecagon Polygon with 15 edges

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.

Tetradecagon Polygon with 14 edges

In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.

Octacontagon Polygon with 80 edges

In geometry, an octacontagon is an eighty-sided polygon. The sum of any octacontagon's interior angles is 14040 degrees.

Hexadecagon Polygon with 16 edges

In mathematics, a hexadecagon is a sixteen-sided polygon.

Hexacontagon Polygon with 60 edges

In geometry, a hexacontagon or hexecontagon or 60-gon is a sixty-sided polygon. The sum of any hexacontagon's interior angles is 10440 degrees.

Tetracontagon Polygon with 40 edges

In geometry, a tetracontagon or tessaracontagon is a forty-sided polygon or 40-gon. The sum of any tetracontagon's interior angles is 6840 degrees.

Icositetragon Polygon with 24 edges

In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.

Enneacontagon Polygon with 90 edges

In geometry, an enneacontagon or enenecontagon or 90-gon is a ninety-sided polygon. The sum of any enneacontagon's interior angles is 15840 degrees.

Hexacontatetragon Polygon with 64 edges

In geometry, a hexacontatetragon or 64-gon is a sixty-four-sided polygon. The sum of any hexacontatetragon's interior angles is 11160 degrees.

Tetracontaoctagon Polygon with 48 edges

In geometry, a tetracontaoctagon or 48-gon is a forty-eight sided polygon. The sum of any tetracontaoctagon's interior angles is 8280 degrees.

Enneacontahexagon Polygon with 96 edges

In geometry, an enneacontahexagon or enneacontakaihexagon or 96-gon is a ninety-six-sided polygon. The sum of any enneacontahexagon's interior angles is 16920 degrees.

120-gon Polygon with 120 edges

In geometry, a 120-gon is a polygon with 120 sides. The sum of any 120-gon's interior angles is 21240 degrees.

360-gon

In geometry, a 360-gon is a polygon with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees.

Icosidigon Polygon with 22 edges

In geometry, an icosidigon or 22-gon is a twenty-two-sided polygon. The sum of any icosidigon's interior angles is 360 degrees.

Icosihexagon Polygon with 26 edges

In geometry, an icosihexagon or 26-gon is a twenty-six-sided polygon. The sum of any icosihexagon's interior angles are 4320°.

Icosioctagon Polygon with 28 edges

In geometry, an icosioctagon or 28-gon is a twenty eight sided polygon. The sum of any icosioctagon's interior angles is 4680 degrees.

Triacontatetragon Polygon with 34 edges

In geometry, a triacontatetragon or triacontakaitetragon is a thirty-four-sided polygon or 34-gon. The sum of any triacontatetragon's interior angles is 5760 degrees.

References

  1. ↑ A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems by Benjamin Franklin Finkel
  2. ↑ Weisstein, Eric W. "Icosidodecagon". MathWorld .
  3. ↑ Constructible Polygon
  4. ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN   978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
  5. ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  6. ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
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